A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X

You write . However, X is the set on which a relation (which one, by the way?) is considered, so there is no reason to assume that X contains pairs. To say that x and y are elements of X, one writes , or . Some pedants even write .

You write . However, X is the set on which a relation (which one, by the way?) is considered, so there is no reason to assume that X contains pairs. To say that x and y are elements of X, one writes , or . Some pedants even write .

That's what I meant. I just forgot to right the square for X.

Let's say we're talking about a set whose elements are ordered pairs.

Could a symmetric relation in a set whose elements are ordered pairs be one that I described above?

A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X

This is like saying, "A subset of a given set X is called nonempty iff there exists an x in X". Since the left-hand side did not introduce a notation for the subset in question ("A subset Y of a given set X..."), the right-hand side cannot talk about this subset, and the whole definition is meaningless. Could you reformulate your definition and collect all relevant info in one post?